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  • Open Access

Robustness of the derivative expansion in asymptotic safety

Alessio Baldazzi1,*, Kevin Falls2,†, Yannick Kluth3,4,‡, and Benjamin Knorr5,§

  • *Contact author: alessio.baldazzi@unitn.it
  • †Contact author: kfalls@fing.edu.uy
  • ‡Contact author: yannick.kluth@manchester.ac.uk
  • §Contact author: knorr@thphys.uni-heidelberg.de
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Phys. Rev. D 113, 026005 – Published 8 January, 2026

DOI: https://doi.org/10.1103/hlrm-d4g2

Abstract

We analyze the renormalization group flow of quantum gravity at sixth order in the derivative expansion within the background field approximation. Nonlinear field redefinitions are used to ensure that only essential couplings flow. Working within the universality class of general relativity, with a vanishing cosmological constant, redundant couplings are fixed to their values at the Gaussian fixed point. This reduces the theory space to two dynamical essential couplings given by Newton’s and the Goroff-Sagnotti coupling. Furthermore, it implements the condition that no extra degrees of freedom are present beyond those of general relativity, in contrast to higher derivative theories and derivative expansions in a conventional renormalization scheme. We find a unique ultraviolet fixed point with a single relevant direction and analyze the phase diagram of the theory. Our results suggest resilience of the gravitational Reuter fixed point under the inclusion of higher order curvature invariants and show several signs of near perturbativity. The regulator dependence of our results is investigated in detail and shows that qualitative and quantitative features are robust to a large extent.

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References (80)

  1. M. H. Goroff and A. Sagnotti, Phys. Lett. 160B, 81 (1985).
  2. M. H. Goroff and A. Sagnotti, Nucl. Phys. B266, 709 (1986).
  3. A. E. van de Ven, Nucl. Phys. B378, 309 (1992).
  4. J. F. Donoghue, Phys. Rev. D 50, 3874 (1994).
  5. S. Weinberg, in Understanding the Fundamental Constitutents of Matter (Springer, Boston, MA, 1978).
  6. S. Weinberg, Ultraviolet divergencies in quantum theories of gravitation, in General Relativity: An Einstein Centenary Survey (Cambridge University Press, Cambridge, England, 1980), pp. 790–831.
  7. G. Parisi, in Cargese Summer Institute: New Developments in Quantum Field Theory and Statistical Mechanics (Springer, Boston, MA, 1977).
  8. M. Reuter, Phys. Rev. D 57, 971 (1998).
  9. B. Knorr, C. Ripken, and F. Saueressig, Form factors in asymptotically safe quantum gravity, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 149.
  10. A. Eichhorn and M. Schiffer, Asymptotic safety of gravity with matter, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 187.
  11. T. R. Morris and D. Stulga, The functional f(R) approximation, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 133.
  12. R. Martini, G. P. Vacca, and O. Zanusso, Perturbative approaches to nonperturbative quantum gravity, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 146.
  13. C. Wetterich, Quantum gravity and scale symmetry in cosmology, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 168.
  14. A. Platania, Black holes in asymptotically safe gravity, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 165.
  15. F. Saueressig, The functional renormalization group in quantum gravity, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 144.
  16. J. M. Pawlowski and M. Reichert, Quantum gravity from dynamical metric fluctuations, in Handbook of Quantum Gravity (Springer Nature, Singapore, 2023), p. 170.
  17. R. Martini, D. Sauro, and O. Zanusso, Phys. Rev. D 109, 065014 (2024).
  18. A. Bonanno, M. Conti, and D. Zappalà, Phys. Lett. B 847, 138311 (2023).
  19. A. M. Bonanno, M. Conti, and S. L. Cacciatori, Phys. Rev. D 108, 026008 (2023).
  20. E. D’Angelo, Phys. Rev. D 109, 066012 (2024).
  21. J. M. Pawlowski and J. Tränkle, Phys. Rev. D 110, 086011 (2024).
  22. A. Eichhorn and S. Ray, Phys. Lett. B 850, 138529 (2024).
  23. A. Bonanno, D. Malafarina, and A. Panassiti, Phys. Rev. Lett. 132, 031401 (2024).
  24. G. P. de Brito, B. Knorr, and M. Schiffer, Phys. Rev. D 108, 026004 (2023).
  25. F. Saueressig and J. Wang, J. High Energy Phys. 09 (2023) 064.
  26. H. Kawai and N. Ohta, Phys. Rev. D 107, 126025 (2023).
  27. B. Knorr, Phys. Rev. D 110, 026001 (2024).
  28. G. P. de Brito, A. Eichhorn, and S. Ray, Phys. Rev. D 112, 046013 (2025).
  29. G. P. de Brito, A. Eichhorn, M. T. Frandsen, M. Rosenlyst, M. E. Thing, and A. F. Vieira, Phys. Rev. D 109, 055022 (2024).
  30. K. Falls, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, arXiv:1301.4191.
  31. K. Falls, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, Phys. Rev. D 93, 104022 (2016).
  32. A. Eichhorn, P. Labus, J. M. Pawlowski, and M. Reichert, SciPost Phys. 5, 031 (2018).
  33. A. Eichhorn, S. Lippoldt, J. M. Pawlowski, M. Reichert, and M. Schiffer, Phys. Lett. B 792, 310 (2019).
  34. K. Falls, N. Ohta, and R. Percacci, Phys. Lett. B 810, 135773 (2020).
  35. B. Knorr, SciPost Phys. Core 4, 020 (2021).
  36. S. Sen, C. Wetterich, and M. Yamada, J. High Energy Phys. 03 (2022) 130.
  37. A. Codello, R. Percacci, and C. Rahmede, Ann. Phys. (Amsterdam) 324, 414 (2009).
  38. D. Benedetti, P. F. Machado, and F. Saueressig, Nucl. Phys. B824, 168 (2010).
  39. H. Gies, B. Knorr, S. Lippoldt, and F. Saueressig, Phys. Rev. Lett. 116, 211302 (2016).
  40. K. Falls, C. R. King, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, Phys. Rev. D 97, 086006 (2018).
  41. Y. Kluth and D. F. Litim, Phys. Rev. D 108, 026005 (2023).
  42. Y. Kluth and D. F. Litim, Phys. Rev. D 106, 106022 (2022).
  43. J. M. Pawlowski, Ann. Phys. (Amsterdam) 322, 2831 (2007).
  44. A. Baldazzi, R. B. A. Zinati, and K. Falls, SciPost Phys. 13, 085 (2022).
  45. A. Baldazzi and K. Falls, Universe 7, 294 (2021).
  46. B. Knorr, arXiv:2204.08564.
  47. K. S. Stelle, Gen. Relativ. Gravit. 9, 353 (1978).
  48. A. Platania and C. Wetterich, Phys. Lett. B 811, 135911 (2020).
  49. A. Platania, J. High Energy Phys. 09 (2022) 167.
  50. C. Wetterich, Phys. Lett. B 301, 90 (1993).
  51. T. R. Morris, Int. J. Mod. Phys. A 09, 2411 (1994).
  52. U. Ellwanger, Z. Phys. C 62, 503 (1994).
  53. F. J. Wegner, J. Phys. C 7, 2098 (1974).
  54. T. R. Morris and Z. H. Slade, J. High Energy Phys. 11 (2015) 094.
  55. R. Percacci and G. P. Vacca, Classical Quantum Gravity 27, 245026 (2010).
  56. N. Ohta, R. Percacci, and G. P. Vacca, Phys. Rev. D 92, 061501 (2015).
  57. N. Ohta, R. Percacci, and G. P. Vacca, Eur. Phys. J. C 76, 46 (2016).
  58. K. Falls, Phys. Rev. D 92, 124057 (2015).
  59. D. Benedetti, Gen. Relativ. Gravit. 48, 68 (2016).
  60. H. W. Hamber and R. Toriumi, Int. J. Mod. Phys. D 22, 1330023 (2013).
  61. B. S. DeWitt, The Global Approach to Quantum Field Theory. Vol. 1, 2 (Oxford University Press, New York, 2003), Vol. 114.
  62. D. F. Litim, Phys. Rev. D 64, 105007 (2001).
  63. J. M. Martin-Garcia, R. Portugal, and L. R. U. Manssur, Comput. Phys. Commun. 177, 640 (2007).
  64. D. Brizuela, J. M. Martin-Garcia, and G. A. Mena Marugan, Gen. Relativ. Gravit. 41, 2415 (2009).
  65. J. M. Martín-García, Comput. Phys. Commun. 179, 597 (2008).
  66. T. Nutma, Comput. Phys. Commun. 185, 1719 (2014).
  67. B. Knorr, Phys. Rev. Lett. 128, 161301 (2022).
  68. J. A. M. Vermaseren, arXiv:math-ph/0010025.
  69. J. Kuipers, T. Ueda, J. A. M. Vermaseren, and J. Vollinga, Comput. Phys. Commun. 184, 1453 (2013).
  70. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hlrm-d4g2 on Mendeley Data; A. Baldazzi, K. Falls, Y. Kluth, and B. Knorr, 10.17632/zfn4rzthcg.1 (2023).
  71. Z. Bern, C. Cheung, H.-H. Chi, S. Davies, L. Dixon, and J. Nohle, Phys. Rev. Lett. 115, 211301 (2015).
  72. Z. Bern, H.-H. Chi, L. Dixon, and A. Edison, Phys. Rev. D 95, 046013 (2017).
  73. T. Papenbrock and C. Wetterich, Z. Phys. C 65, 519 (1995).
  74. A. Baldazzi, R. Percacci, and L. Zambelli, Phys. Rev. D 103, 076012 (2021).
  75. A. Baldazzi, R. Percacci, and L. Zambelli, Phys. Rev. D 104, 076026 (2021).
  76. I. Balog, H. Chaté, B. Delamotte, M. Marohnic, and N. Wschebor, Phys. Rev. Lett. 123, 240604 (2019).
  77. T. Denz, J. M. Pawlowski, and M. Reichert, Eur. Phys. J. C 78, 336 (2018).
  78. A. Baldazzi, K. Falls, and R. Ferrero, Ann. Phys. (Amsterdam) 440, 168822 (2022).
  79. K. Falls, Eur. Phys. J. C 81, 121 (2021).
  80. D. Buccio and R. Percacci, J. High Energy Phys. 10 (2022) 113.

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Phys. Rev. D 113, 026005– Published 8 January, 2026
  • Received 26 February 2024
  • Accepted 16 October 2025
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